LIN1032 Formal Foundations for Linguistics

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1 LIN1032 Formal Foundations for Lecture 1 Albert Gatt

2 Practical stuff Course tutors: Albert Gatt (first half) Ray Fabri (second half) Course website: TBA

3 Practical stuff Suggested course text: J. Allwood, L-G Andersson, O. Dahl (1977). Logic in. Cambridge University Press Assessment: end of semester test Structure of lectures: a lot of practical work, in addition to explanation!

4 The subject matter In many areas of linguistics, we use mathematical techniques. Some examples: trees in syntactic description the use of logic in analysing the meaning of sentences the use of sets to define the things denoted by a word

5 What is Mathematics, really? It s not just about numbers! Mathematics is much more than that: Mathematics is, most generally, the study of any and all certain truths (about any kind of concepts) These concepts can be about numbers, symbols, ideas, images, sounds, anything!

6 So, what s this class about? Our focus is on discrete mathematical structures Discrete ( discreet!) Composed of distinct, separable parts. Opposite of continuous. Structures Objects built up from simpler objects. Discrete Mathematics The mathematical study of discrete objects and structures.

7 Uses of Discrete Math Starting from simple structures of logic and set theory, theories are constructed that capture aspects of reality: Physics Biology (DNA) Common-sense reasoning (logic) Natural Language (trees, sets, functions,..) Anything that we want to describe precisely

8 Aims of the course This course aims to introduce some of the discrete mathematical methods which are useful in various areas of linguistics. The aim is introductory. We are not interested in becoming mathematicians, but in becoming informed consumers of certain of these techniques.

9 Course contents Five main parts: 1. Set theory 2. Propositional logic 3. Predicate logic 4. Trees and graphs 5. Feature structures

10 Course contents Five main parts: 1. Set theory we ll start here 2. Propositional logic 3. Predicate logic 4. Trees and graphs 5. Feature structures

11 Part 1 The concept of a set

12 Introduction to Set Theory A set is a type of structure, representing an unordered collection of zero or more distinct objects. sometimes referred to as a collection or an aggregate Set theory deals with operations between, relations among, and statements about sets.

13 The concept of a set You can define a set of anything. The members of a set do not need to have something in common. We re usually interested in sets whose members do have some meaningful relationship

14 Why sets? Possibly the most fundamental mathematical structure: All of mathematics can be defined in terms of some form of set theory. More important for us: set theory is used in much of formal semantics e.g. we can take the meaning of the word dog to be a certain set of things e.g. the meaning of love can be characterised as a relation between members of two sets many other structures, such as trees, can be defined in terms of sets

15 Some examples of sets the set of all winners of the Nobel Prize for Literature = {William Faulkner, Albert Camus, Gabriel Garcia Marquez, } this set is finite the set of all the positive integers = {1, 2, 3, 4, } this set is infinite the set of all students

16 Intuition behind sets Almost anything you can do with individual objects, you can also do with sets of objects. E.g. (informally speaking), you can refer to them, compare them, combine them, You can also do some things to a set that you probably cannot do to an individual: E.g., you can check whether one set is contained in another determine how many elements it has

17 Venn diagrams for representing sets donkey girl sky house ship shape rat robber river red ripping rode the set of all English nouns these sets have something in common the set of English words beginning with r

18 Exercise 1 Assume we have a small universe consisting of: Lidwina, who studies and Mathematics Steve, who studies Mathematics êensu, who studies Philosophy, Mathematics and Stephanie, who studies Olga, who studies Philosophy Draw a Venn diagram to represent these students according to the courses they study.

19 Exercise 1 Mathematics Stephanie Lidwina Steve êensu Philosophy Olga

20 Exercise 1 (continued) Indicate, by shading the right parts of the Venn diagram, all the parts which consist of students who study only, or Philosophy only, or both and Philosophy but not Mathematics

21 Exercise 1 Mathematics Stephanie Lidwina Steve êensu Philosophy Olga These parts do not belong to the shaded area

22 Basic notation for sets Venn Diagrams are useful, but sometimes a bit cumbersome. For sets, we ll use variables S, T, U, by convention, set variables are in uppercase

23 Defining a set I We can define a set by enumeration i.e. by listing all its elements or members by convention, we use curly brackets {} L L = {Stephanie, Lidwina, êensu} Stephanie Lidwina êensu

24 Defining a set II Another way of defining a set is by description L Stephanie Lidwina êensu L = {x x is a student of } In English: L is the set of all those x, such that x is a student of. NB: In this notation, we use lowercase x to denote a variable. In our notation, x ranges over students of.

25 Definition by description Defining a set by description is much more flexible than by enumeration. We can even define an infinite set N = {x x is a natural number} By enumeration: N = {0, 1, 2, 3, 4, 5 }

26 Exercise 2 Define the set of integers (whole numbers) between 1 and 5 by enumeration. {1,2,3,4,5} Define the following by description: the set of all things which are red {x x is red} the set of all women with a moustache {x x is a woman and x has a moustache} the set of all men who are bald {x x is a man and x is bald} Be as explicit as possible: better x is a man and x is bald than x is a bald man

27 Basic properties of sets Sets are inherently unordered: No matter what objects a, b, and c denote, {a, b, c} = {a, c, b} = {b, a, c} = Multiple listings make no difference: {a, a, c, c, c, c}={a,c} This is why our definition stated that a set is an unordered collection of distinct elements.

28 Definition of Set Equality Two sets are equal if and only if they contain exactly the same elements. It does not matter how the set is defined For example: {1, 2, 3, 4} = {x x is an integer where x is greater than 0 and x is less than 5 }

29 The Principle of Extensionality This states that: two sets A and B are distinct (i.e. not equal) if, and only if, there is at least one element that A and B do not have in common Example: A = {a,b,c,d} B = {a,b,c} A B

30 Basic Set Relations: element of x S ( x is in S ) object x is an lement or member of set S x S object x is not an element or member of set S

31 Example: element of relation L Stephanie Lidwina Steve M êensu P Olga Stephanie L Olga P Steve P

32 Subset relation S T ( S is a subset of T ) every element of S is also an element of T NB: This means that if S = T, then S T Example: A = {a,b,c,d} B = {b,c} B A

33 Superset relations S T ( S is a superset of T ) Every element of T is also an element of S So, S T iff T S

34 Some more on equality Since S and T are equal if they have exactly the same elements, we have that: S = T iff S T and T S

35 Subset/superset: illustration Assume that: S = {x x is a student} P = {y y is a student and y studies philosophy} S P Notation: S P P S

36 Proper (Strict) Subsets & Supersets S T ( S is a proper subset of T ) T S S T but. Example:{1,2} {1,2,3} We have {1,2,3} {1,2,3}, but not {1,2,3} {1,2,3}

37 Subset/superset: illustration Assume that: S = {x x is a student} P = {y y is a student and y studies philosophy} S P Notation: S P P S But also: P S

38 Summary of Notation A, B, : uppercase letters used for sets a B: a is an element of B A B: A is a subset of B (A may also be equal to B) A B: A is a superset of B (Again, A may also be equal to B) A B: A is a proper subset of B (A is NOT equal to B)

39 Exercise 3 Express these in symbols b is an element of A b A C is a subset of B C B A is not equal to D A D C is a proper subset of F C F

40 Part 2 Some special kinds of sets

41 Singleton sets and empty sets A set with only one element is called a singleton set sometimes referred to as a unit set Sets can also be empty. Examples of empty sets: the set of dogs that can program computers the set of female presidents of the United States

42 Some more on the empty set Recall the principle of extensionality: two sets are distinct if, and only if, they have at least one element which is in one but not in the other but this means that if two sets are empty, they must be the same set there is no element which is in one but not in the other In set theory, we say there is only one empty set ( the empty set). The empty set is a subset of every set.

43 Notation for the empty set We have seen that there exists exactly one empty set, so we can give it a name: ( the empty set ) is the unique set that contains no elements whatsoever. = {} This has a rather strange consequence: the set of dogs that can program computers = the set of female presidents of the United States

44 Cardinality and Finiteness S (read the cardinality of S ) is a measure of how many different elements S has. =0 {1,2,3} = 3 {a,b} = 2

45 Infinite Sets Sets may be infinite (i.e., not finite, without end, unending). Symbols for some special infinite sets: N = {0, 1, 2, } The Natural numbers. Z = {, -2, -1, 0, 1, 2, } The integers. R = The Real numbers, such as Blackboard Bold or double-struck font (N,Z,R) is also often used for these special number sets.

46 The Power Set Operation The power set P(S) of a set S is the set of all subsets of S. P(S) = {x x S}. E.g. P({a,b}) = {, {a}, {b}, {a,b}} NB: The set itself is a member of its own power set! NB: The empty set is always in the power set! Notice that the power set is a set of sets. Sets too can be elements of sets!

47 Exercise 4 Let A = {Edinburgh, London, Dublin} What is P(A)? {, {Edinburgh}, {London}, {Dublin}, {Edinburgh,London}, {Edinburgh,Dublin}, {London, Dublin}, {Edinburgh, Dublin, London}} What is P(A)? Observe that P(A) = 2 A

48 Summary We ve introduced one of the core discrete mathematical structures. Also some of the core relations on sets: element of subset/superset strict subset Things to remember: The Principle of Extensionality The empty set as a special case: there is only one. The powerset operation

49 Acknowledgements Some of these materials are adapted from slides by Kees van Deemter (University of Aberdeen), and Michael Frank (University of Florida)

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